Preconditioning of elliptic problems by approximation in the transform domain

Abstract

Preconditioned conjugate gradient method is applied for solving linear systems Ax = b where the matrix A is the discretization matrix of second-order elliptic operators. In this paper, we consider the construction of the transform based preconditioner from the viewpoint of image compression. Given a smooth image, a major portion of the energy is concentrated in the low frequency regions after image transformation. We can view the matrix A as an image and construct the transform based preconditioner by using the low frequency components of the transformed matrix. It is our hope that the smooth coefficients of the given elliptic operator can be approximated well by the low-rank matrix. Numerical results are reported to show the effectiveness of the preconditioning strategy. Some theoretical results about the properties of our proposed preconditioners and the condition number of the preconditioned matrices are discussed.